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Response surface methodology

response surface methodology, response surface methodology pdf
In statistics, response surface methodology RSM explores the relationships between several explanatory variables and one or more response variables The method was introduced by George E P Box and K B Wilson in 1951 The main idea of RSM is to use a sequence of designed experiments to obtain an optimal response Box and Wilson suggest using a second-degree polynomial model to do this They acknowledge that this model is only an approximation, but they use it because such a model is easy to estimate and apply, even when little is known about the process

Statistical approaches such as RSM can be employed to maximize the production of a special substance by optimization of operational factors In contrast to conventional methods, the interaction among process variables can be determined by statistical techniques


  • 1 Basic approach of response surface methodology
  • 2 Important RSM properties and features
  • 3 Special geometries
    • 31 Cube
    • 32 Sphere
    • 33 Simplex geometry and mixture experiments
  • 4 Extensions
    • 41 Multiple objective functions
  • 5 Practical concerns
  • 6 See also
  • 7 References
    • 71 Historical
  • 8 External links

Basic approach of response surface methodology

An easy way to estimate a first-degree polynomial model is to use a factorial experiment or a fractional factorial design This is sufficient to determine which explanatory variables affect the response variables of interest Once it is suspected that only significant explanatory variables are left, then a more complicated design, such as a central composite design can be implemented to estimate a second-degree polynomial model, which is still only an approximation at best However, the second-degree model can be used to optimize maximize, minimize, or attain a specific target for

Important RSM properties and features


ORTHOGONALITY: The property that allows individual effects of the k-factors to be estimated independently without or with minimal confounding Also orthogonality provides minimum variance estimates of the model coefficient so that they are uncorrelated

ROTATABILITY: The property of rotating points of the design about the center of the factor space The moments of the distribution of the design points are constant

UNIFORMITY: A third property of CCD designs used to control the number of center points is uniform precision or Uniformity

Special geometries


Cubic designs are discussed by Kiefer, by Atkinson, Donev, and Tobias and by Hardin and Sloane


Spherical designs are discussed by Kiefer and by Hardin and Sloane

Simplex geometry and mixture experiments

Mixture experiments are discussed in many books on the design of experiments, and in the response-surface methodology textbooks of Box and Draper and of Atkinson, Donev and Tobias An extensive discussion and survey appears in the advanced textbook by John Cornell


Multiple objective functions

See also: Multiobjective optimization and Pareto efficiency

Some extensions of response surface methodology deal with the multiple response problem Multiple response variables create difficulty because what is optimal for one response may not be optimal for other responses Other extensions are used to reduce variability in a single response while targeting a specific value, or attaining a near maximum or minimum while preventing variability in that response from getting too large

Practical concerns

Response surface methodology uses statistical models, and therefore practitioners need to be aware that even the best statistical model is an approximation to reality In practice, both the models and the parameter values are unknown, and subject to uncertainty on top of ignorance Of course, an estimated optimum point need not be optimum in reality, because of the errors of the estimates and of the inadequacies of the model

Nonetheless, response surface methodology has an effective track-record of helping researchers improve products and services: For example, Box's original response-surface modeling enabled chemical engineers to improve a process that had been stuck at a saddle-point for years The engineers had not been able to afford to fit a cubic three-level design to estimate a quadratic model, and their biased linear-models estimated the gradient to be zero Box's design reduced the costs of experimentation so that a quadratic model could be fit, which led to a long-sought ascent direction

See also

  • Plackett–Burman design
  • Box–Behnken design
  • Central composite design
  • IOSO method based on response-surface methodology
  • Optimal designs
  • Polynomial regression
  • Polynomial and rational function modeling
  • Surrogate model
  • Probabilistic design
  • Gradient-Enhanced Kriging GEK


  1. ^ Asadi, Nooshin; Zilouei, Hamid March 2017 "Optimization of organosolv pretreatment of rice straw for enhanced biohydrogen production using Enterobacter aerogenes" Bioresource Technology 227: 335–344 doi:101016/jbiortech201612073 
  2. ^ Ahmed Maged, Salah Haridy, Mohammad Shamsuzzaman, Imad Alsyouf, and Roubi Zaied 2018Statistical Monitoring and Optimization of Electrochemical Machining using Shewhart Charts and Response Surface Methodology32, 68-77 https://doiorg/1026776/ijemm0302201801
  3. ^ Box, G E P and Wilson, KB 1951 On the Experimental Attainment of Optimum Conditions with discussion Journal of the Royal Statistical Society Series B131:1–45
  4. ^ Improving Almost Anything: Ideas and Essays, Revised Edition Wiley Series in Probability and Statistics George E P Box
  5. ^ Soltani, M and Soltani, J 2016 Determination of optimal combination of applied water and nitrogen for potato yield using response surface methodology RSM Journal of Bioscience Biotechnology Research Communication 91: 46-54 Online Contents Available at: http://wwwbbrcin
  • Box, G E P and Wilson, KB 1951 On the Experimental Attainment of Optimum Conditions with discussion Journal of the Royal Statistical Society Series B 131:1–45
  • Box, G E P and Draper, Norman 2007 Response Surfaces, Mixtures, and Ridge Analyses, Second Edition , Wiley
  • Atkinson, A C and Donev, A N and Tobias, R D 2007 Optimum Experimental Designs, with SAS Oxford University Press pp 511+xvi ISBN 978-0-19-929660-6  External link in |publisher= helpCS1 maint: Multiple names: authors list link
  • Cornell, John 2002 Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data third ed Wiley ISBN 0-471-07916-2 
  • Goos, Peter 2002 The Optimal Design of Blocked and Split-plot Experiments Lecture Notes in Statistics 164 Springer  External link in |publisher= help
  • Kiefer, Jack Carl 1985 L D Brown; et al, eds Jack Carl Kiefer Collected Papers III Design of Experiments Springer-Verlag ISBN 0-387-96004-X 
  • Pukelsheim, Friedrich 2006 Optimal Design of Experiments SIAM ISBN 978-0-89871-604-7  External link in |publisher= help
  • R H Hardin and N J A Sloane, "A New Approach to the Construction of Optimal Designs", Journal of Statistical Planning and Inference, vol 37, 1993, pp 339-369
  • R H Hardin and N J A Sloane, "Computer-Generated Minimal and Larger Response Surface Designs: I The Sphere"
  • R H Hardin and N J A Sloane, "Computer-Generated Minimal and Larger Response Surface Designs: II The Cube"
  • Ghosh, S; Rao, C R, eds 1996 Design and Analysis of Experiments Handbook of Statistics 13 North-Holland ISBN 0-444-82061-2 
    • Draper, Norman & Lin, Dennis K J "Response Surface Designs" pp 343–375  Missing or empty |title= help
    • Gaffke, N & Heiligers, B 1996 "Polynomial Regression" Handbook of Statistics, Volume 13 Design and Analysis of Experiments pp 1149–1199 doi:101016/S0169-71619613032-7 


  • Gergonne, J D 1974 "The application of the method of least squares to the interpolation of sequences" Historia Mathematica Translated by Ralph St John and S M Stigler from the 1815 French ed 1 4: 439–447 doi:101016/0315-08607490034-2 
  • Stigler, Stephen M 1974 "Gergonne's 1815 paper on the design and analysis of polynomial regression experiments" Historia Mathematica 1 4: 431–439 doi:101016/0315-08607490033-0 
  • Peirce, C S 1876 "Note on the Theory of the Economy of Research" Coast Survey Report: 197–201  Appendix No 14 NOAA PDF Eprint Reprinted in Collected Papers of Charles Sanders Peirce 7 1958  paragraphs 139–157, and in Peirce, C S July–August 1967 "Note on the Theory of the Economy of Research" Operations Research 15 4: 643–648 doi:101287/opre154643  Abstract at JSTOR
  • Smith, Kirstine 1918 "On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations" Biometrika 12 1/2: 1–85 doi:102307/2331929 JSTOR 2331929 

External links

  • Response surface designs
  • Response surface analysis

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